The Neoclassical Theory of the Firm
Under Moral Hazard
Michael T. Rauh
Indiana University
[email protected]
April 6, 2017
Abstract
We develop a neoclassical theory of the firm under moral hazard with endogenous employment,
endogenous capital, and an external competitive labor market. The crucial assumptions are
that effort becomes harder to measure as employment grows and the exogenous parameters are
affiliated. The model explains why incentives decline but wages rise with firm size, the mixed
evidence on the risk-reward tradeoff, and the positive correlation between wages and profits. In
the long run there is a positive relationship between incentives and risk driven by endogenous
capital. Finally, the model makes novel predictions about the relationship between incentives
and aggregate labor market conditions.
JEL Classifications: D02, D21, D86, J31, M52.
Keywords: employment, incentives, moral hazard, size-wage differential, rent-sharing, risk-reward
tradeoff, wages.
1
Introduction
In this paper, we develop a neoclassical theory of the firm under moral hazard with endogenous
employment, endogenous capital, and an external perfectly competitive labor market.1 Our purpose
is twofold. First, we show that the model is capable of explaining the following important stylized
facts about incentives and wages:
(SF1) Incentives decline with the size of the firm as measured by either employment or capital.
Bishop (1987); Rasmusen and Zenger (1990); Garen (1994); Schaefer (1998); Zenger and
Marshall (2000); and Zenger and Lazzarini (2004).2
(SF2) The classical risk-reward tradeoff in the principal-agent literature implies that incentives
should decline with risk but the evidence as surveyed in Prendergast (2002a) and Devaro
and Kurtulus (2010) is mixed. Indeed, a majority of studies find a positive relationship
between incentives and risk.
(SF3) The size-wage differential, which states that wages increase with firm size. Barron, Black,
and Loewenstein (1987); Brown and Medoff (1989); Abowd, Kramarz, and Margolis (1999);
Troske (1999); and many others.
(SF4) Rent-sharing, as manifested by a positive correlation between wages and profits. Blanchflower, Oswald, and Sanfey (1996); Arai (2003); and several others.
The model is an extension of Holmström and Milgrom (1987), which assumes linear contracts,
exponential utility, and a normally distributed additive shock. In the short run, the firm can freely
vary employment but the capital stock is fixed. The endogenous variables are therefore incentives,
wages, employment, and profit while the exogenous parameters are price, total factor productivity,
capital, the inverse of capital costs, and the inverse of risk. We specify the latter two parameters in
terms of their inverses so we can state that optimal employment is increasing in all the parameters.
We make two crucial assumptions. The first is that effort becomes harder to measure as employment
grows, an idea that dates back at least to Stigler (1962) and has been formally analyzed in Garen
(1985); Ziv (1993); Auriol, Friebel, and Pechlivanos (1999, 2002); Liang, Rajan, and Ray (2008);
1
We use the expression “theory of the firm” in the sense that it is used in microeconomics textbooks rather than
the literature on the theory of the firm which considers a choice between institutions, usually the firm or the market.
The model is “neoclassical” in the usual sense that the firm operates in perfectly competitive capital, labor, and
product markets.
2
There is some evidence that larger firms are more likely to use explicit incentive mechanisms such as piece rates;
e.g., see Brown and Medoff (1989) and Devaro and Kurtulus (2010). This does not necessarily conflict with evidence
that incentives are stronger in smaller firms, where incentives may be implicit.
1
and Rauh (2014).3 The second main assumption, following Holmström and Milgrom (1994), is that
the parameters are affiliated, so a firm which has a high value for one parameter will tend to have
high values for all of them. Since optimal employment is increasing in all the parameters, such a
firm will tend to be a large employer which offers weak incentives because effort is hard to measure
in large firms (SF1) but high wages as a compensating differential for risk (SF3). Since workers are
compensated on the basis of output, which under certain conditions is correlated with profit, wages
will be positively correlated with profit (SF4). In our model, what appears to be rent-sharing is
actually a consequence of performance-related pay.
A substantial theoretical literature already exists for (SF2)-(SF4) and we discuss each of them
more fully below. What sets our model apart from the existing theoretical literatures on the sizewage differential and rent-sharing is that most current explanations are based on external market
considerations, whereas in this paper we offer a complementary internal explanation based on
incentive contracting within the firm. Furthermore, most of these explanations rely on differences
across workers or the existence of market power in product and/or labor markets, whereas in this
paper we show that a size-wage differential and apparent risk-sharing can arise even when workers
are identical and all markets are perfectly competitive. An important advantage of our approach
is that it provides a comprehensive explanation for all four stylized facts within a single theoretical
framework.4
A concise summary of these theoretical results is provided by the classic empirical paper by
Dickens and Katz (1986). Their paper seeks to explain substantial differences in wages across
industries for otherwise seemingly identical workers and to identify those industry characteristics
which explain these wage differences. Summarizing their findings from many different specifications,
they conclude that three explanatory variables stand out: education (corresponding to total factor
productivity), the capital-labor ratio, and the profit variables. The main obstacle to obtaining
robust results is that the explanatory variables are highly correlated. After performing a principal
component analysis, the authors find that the first factor accounts for over a third of the variance
of the industry variables. This is the empirical expression of our assumption that the exogenous
variables are affiliated and therefore tend to move together. The list of variables which are correlated
with this one dominant factor includes wages, variables related to total factor productivity (labor
3
This premise seems widely accepted in the literature. The only concrete evidence we are aware of is Troske
(1999), who finds that the number of supervisors per employee is uncorrelated with both establishment and firm size.
This implies that the number of supervisors increases proportionately with employment.
4
Of course, there are other stylized facts the model cannot explain, such as the fact that larger firms have lower
quit rates (e.g., see Bertola and Garibaldi (2001)).
2
productivity, education, and job tenure), firm and establishment size in terms of employment, the
capital-labor ratio, and the profit variables. This is a rather striking empirical portrait of our
theoretical results.
The other major objective of the paper is to provide a simple theoretical framework which
incorporates moral hazard but has the same broad scope of application as the textbook competitive
model. In particular, situating the firm within a competitive labor market allows us to move beyond
the above stylized facts to make new predictions about the relationship between incentives and
aggregate labor market conditions:
(P1) A leftward shift in the supply of labor (e.g., a reduction in labor force participation) will lead
to stronger incentives at all firms.
(P2) We show that the short-run optimal incentive does not depend on firm risk (SF2), but a
leftward shift in the aggregate demand for labor due to a market-wide increase in risk leads
to weaker incentives at all firms. In other words, there is no observable risk-reward tradeoff
at the firm level but there is one at the market level.
More generally, we show that the optimal incentive is increasing in the workers’ outside option
which is the highest alternative total payoff available in the labor market. Since the participation
constraint binds, the total payoff of each worker is equal to the outside option which is determined
by supply and demand in the labor market. Intuitively, an increase in the outside option (due
to a rightward shift in demand or a leftward shift in supply) increases the cost of employment.
The subsequent reduction in employment makes effort easier to measure so firms offer stronger
incentives. Predictions (P1) and (P2) above are special cases of this general argument. In the
principal-agent and teams literatures, the outside option is usually normalized to zero because all
it does is determine the distribution of the surplus between the principal and agent(s). In contrast,
in our model with endogenous employment the outside option is an important endogenous variable
which firms take as given when choosing employment, incentives, and wages. Garen (1994) provides
some preliminary evidence in support of (P2) but we are unaware of any evidence on (P1).
In the long run, firms are free to adjust their capital stocks and we can add capital to the previous
list of endogenous variables. The fact that firms in our model make explicit capital decisions leads
to further new predictions.
(P3) An increase in the cost of capital will lead to stronger incentives.
(P4) In the long run there is a positive relationship between incentives and risk.
3
Prediction (P4) may seem counterintuitive from the standpoint of conventional contract theory but
is consistent with some evidence as mentioned in the context of (SF2). In our model, an increase
in the cost of capital (P3) or an increase in risk (P4) reduces the demand for capital which in turn
reduces the demand for labor because employment and capital are complements in the production
function. Again, the reduction in employment makes effort easier to measure and leads to stronger
incentives. In both cases, the initial impact on incentives operates through the endogenous capital
stock so these are inherently long-run effects.
In the formal model we follow Holmström (1982) and assume incentive contracts based on group
performance and show that contracts based instead on individual performance do not necessarily
change the results. In fact, the direct nature of the intuitions suggests that our results should hold
in any model where earnings depend on measured performance and measuring performance gets
more difficult as employment increases. This includes not only explicit incentives tied to individual
or group performance but also implicit incentives and perhaps other incentive mechanisms such as
promotion tournaments.
The plan for the rest of the paper is as follows. In the next section we lay out the model
primitives. In section 3, we consider the short run where the capital stock is fixed. We consider
the long run in section 4. Section 5 concludes.
2
Model Primitives
We consider a labor market where the supply side consists of infinitely many identical workers.
The demand side consists of n firms, where n is large enough to justify perfect competition in the
labor market. These firms all hire labor from the same labor pool but may operate in different
product markets, producing different outputs with different production technologies. All markets
are perfectly competitive but we only explicitly model equilibrium in the labor market.
We now describe the production function. Let Li be the number of workers employed by firm
i. Each worker j at firm i chooses an effort level eij (we discuss this choice below) and experiences
an idiosyncratic random shock ij . Total effort at firm i is given by
Ei =
Li
X
eij
(1)
ij .
(2)
j=1
and the firm shock by
i =
Li
X
j=1
4
For each firm i the random variables {ij } are i.i.d. normal across workers with mean zero and
variance σi2 , so the firm shock i is normal with mean zero and variance σi2 Li . Note that greater
employment Li increases the volatility of the firm shock. Since workers are identical, the variance
σi2 is the same for all workers but may vary across firms depending on their production technology.
The stochastic production function of firm i is given by
qi = Ai f (Ki )Ei + i ,
(3)
where qi is output, Ai > 0 is total factor productivity, Ki the capital stock, and f satisfies f (0) = 0,
f 0 > 0, and f 00 < 0. The deterministic part of (3) is essentially Cobb-Douglas in capital Ki and
total effort Ei . The total factor productivity parameter Ai captures investments in human capital,
process improvements such as employee involvement programs, and broader aspects of technological
progress such as specialization and division of labor (none of which are modeled here). The function
f reflects the productivity of capital. The stochastic part i represents the crucial assumption that
each additional worker adds an additional productivity shock. For example, consider an assembly
line where a separate workstation is created for each new worker. This adds one more stage in the
production process and therefore one more source of uncertainty. Another interpretation, discussed
below, is that each additional worker adds additional measurement error in group performance
measurement.
We assume moral hazard in the sense that the firm observes output qi but not total Ei or
individual efforts. As the size of the firm in terms of employment Li grows, the variance of output
σi2 Li increases and output becomes a noisier signal of effort. In this sense, effort becomes harder
to measure. We refer to σi2 as the marginal variance because it represents the increase in the
variance of output due to the marginal hire. The parameters Ai and σi2 characterize the production
technology of firm i and summarize technological differences across firms.
For most of the paper we follow Holmström (1982) and assume that individual performance
measures are not available and that output (3) is the only contractible performance measure.5 In
particular, firm i offers a linear contract in output
Ii = αi + βi qi = αi + βi Ai f (Ki )Ei + βi i ,
(4)
where Ii is the wage or income, αi the fixed component of compensation, and βi the incentive
5
A performance measure is contractible if it is observable to the parties to the contract and verifiable to third
parties such as the courts. It can therefore be included as part of an enforceable contract.
5
parameter.6 We discuss individual performance measures briefly below. Note that incentives tied
to stochastic output expose workers to risk. In particular, the effect on income (4) of fluctuations
in the firm shock i is magnified by the incentive βi as is evident from the final term.
Signal Interpretation
In the above interpretation of the model, workers are compensated based on actual output (3)
which is subject to productivity shocks i . We call this the output interpretation of the model.
Alternatively, we could assume that output is deterministic
qi = Ai f (Ki )Ei
(5)
but that firms must pay their workers before output is realized based on a noisy signal of output
yi = qi + i .
(6)
In this interpretation of the model, i represents noise, so each additional hire increases noise or
measurement error. The corresponding contract is
Ii = αi + βi yi .
(7)
We call this the signal interpretation of the model. These two interpretations of the model are
obviously equivalent but their empirical implications differ. In the output version, a reduction in
the marginal variance σi2 can be interpreted in terms of quality control or process improvements,
whereas in the signal version it can be interpreted as an increase in monitoring which improves the
informativeness of the signal. For concreteness, we proceed with the output interpretation because
it involves less notation. Let Mi be the subset of the n firms which operates in the same product market as firm i and
pMi the perfectly competitive price determined by supply and demand in that market. Let ρi be
the cost of capital, which reflects the cost of borrowing for firm i. Since firms can differ in terms of
6
As is well-known (see Bolton and Dewatripont (2005, Section 4.3)), linear contracts are generally suboptimal in
the context of additive and normally distributed productivity shocks and a single agent with CARA utility. Given
the structure of our model, in particular the additive nature of the production function, there is reason to believe
that the classic rationale for linear contracts provided by Holmström and Milgrom (1987) for the single-agent context
also applies here. Under limited liability, Bose, Pal, and Sappington (2011) show that the optimal linear contract
achieves at least 90% of the expected profit of the optimal nonlinear contract. A linear contract will therefore be
optimal when transaction costs are sufficiently increasing in contractual complexity.
6
price pMi , total factor productivity Ai , capital Ki , and marginal variance σi2 , their creditworthiness
and hence their borrowing costs will also generally differ. The expected wage wi and wage Ii are
given by
wi = αi + βi Ai f (Ki )Ei
(8)
Ii = wi + βi i
(9)
and profit Π̃i and expected profit Πi by
Π̃i = pMi qi − Ii Li − ρi Ki
(10)
Πi = pMi Ai f (Ki )Ei − wi Li − ρi Ki .
(11)
Workers have the CARA utility function
− exp −r Ii − (1/2)e2ij ,
(12)
where r is the CARA coefficient and (1/2)e2ij the cost of effort. A higher value of r indicates that
workers are more risk averse. A worker’s payoff at firm i is given by7
Ui = αi + βi Ai f (Ki )Ei − (1/2)e2ij − RPi ,
(13)
RPi = (1/2)rβi2 σi2 Li
(14)
where
is the workers’ risk premium which represents the disutility of risk. This is essentially the variance
of income βi2 σi2 Li scaled by the workers’ degree of risk aversion r. The risk premium is increasing
in the incentive βi because stronger incentives increase the variance of income. Let ui be the best
alternative payoff for workers at firm i across the other n − 1 firms which participate in this labor
market
ui = max Uk .
k6=i
(15)
Perfect competition will ensure that all firms will offer the same total payoff Ui so we usually write
u without the subscript.
7
For a derivation, see Bolton and Dewatripont (2005, p. 137). Since all workers at firm i choose the same effort
level eij , we write Ui instead of Uij .
7
3
The Short Run
The short run is defined as that period of time when firms’ capital stocks are fixed. We first
consider the workers’ optimal choice of effort. Each employed worker j at firm i chooses effort eij
to maximize her payoff (13). The solution is
ei = βi Ai f (Ki )
(16)
for all workers j, so we drop the unnecessary subscript j. As usual, stronger incentives βi inspire
greater effort because they increase the reward for additional output. But incentives are not the
only factor which determines effort. An increase in total factor productivity Ai or the capital stock
Ki also inspires greater effort because they increase the effectiveness of effort in generating output,
which is the basis for rewards. Furthermore, an increase in Ai or Ki increases the sensitivity of
effort to incentives. Note that these effects of Ai and Ki are conditional on the existence of some
form of incentive pay. Otherwise, an increase in the expected marginal product of effort would not
induce greater effort because output is not rewarded.
The short-run problem of firm i is to choose the contract (αi , βi ) and employment level Li ≥ 0
to maximize expected profit (11) subject to two constraints: the incentive compatibility constraint
(16) and the participation constraint Ui ≥ u. The incentive compatibility constraint captures the
moral hazard problem internal to the firm while the participation constraint reflects the external
competitive pressure on internal incentive contracting. It is clear that the firm will choose αi to
make the participation constraint bind. Substituting Ui = u or
wi = (1/2)e2i + (1/2)rβi2 σi2 Li + u
(17)
Πi = pMi Ai f (Ki )Ei − (1/2)e2i + (1/2)rβi2 σi2 Li + u Li − ρi Ki .
(18)
into (11),
The participation constraint (17) implies that workers must be compensated for an increase in
effort, the risk premium (14), or the outside option u. Substituting optimal effort (16),
Πi = pMi A2i f (Ki )2 βi Li − (1/2)A2i βi2 f (Ki )2 + (1/2)rβi2 σi2 Li + u Li − ρi Ki .
(19)
We now derive the optimal incentive βi for a given employment level Li . From (19), the expected
benefit of incentives is that they increase expected revenue by increasing effort and expected output
8
(the first term). The cost is that incentives necessitate an increase in the expected wage wi in (17)
because of the higher cost of effort and the increase in the risk premium (the second and third
terms). The optimal βi for a given employment level Li balances these tradeoffs
βi =
pMi A2i f (Ki )2
.
A2i f (Ki )2 + rσi2 Li
(20)
In comparison, the optimal incentive in Milgrom and Roberts (1992, p. 221) in a single agent
context is given by
β=
P 0 (e)
,
1 + rσ 2
(21)
where r and σ 2 are the same as in this paper and P (e) is the expected benefit of effort to the
principal. Since P (e) presumably captures such effects as the price pMi and productivity Ai , our
expression (20) has a similar structure except that it also depends on capital Ki and employment
Li . But βi in (20) is not the optimal incentive in our model because employment is endogenous.
After substituting the optimal employment level into (20), it will lose the familiar structure in (21).
In our model, an increase in the price pMi makes effort more valuable to the firm, which responds
with stronger incentives βi . An increase in the CARA coefficient r or the variance σi2 Li of output
increases the cost of incentives in terms of the risk premium which leads to weaker incentives. This
is a version of the classical risk-reward tradeoff already evident in (21). In particular, an increase
in employment Li leads to weaker incentives by making effort harder to measure. Incentives are
increasing in total factor productivity Ai and the capital stock Ki due to two separate effects.
First, an increase in Ai or Ki makes effort more valuable to the firm. Second, an increase in Ai
or Ki increases the sensitivity of effort to incentives in the incentive compatibility constraint (16).
These two effects explain why Ai and Ki enter (20) as squared terms. Note that the effect of firm
size on incentives is currently ambiguous: an increase in firm size as measured by employment
Li leads to weaker incentives whereas an increase in firm size as measured by assets Ki leads to
stronger incentives. Once the endogeneity of employment Li is taken into account, we will find
that the optimal incentive is unambiguously decreasing in firm size as measured by either capital
or employment.
3.1
The Demand for Labor and Equilibrium in the Labor Market
As we have seen, each firm i takes the workers’ outside option u as given and chooses the fixed
component of pay αi to make the participation constraint bind Ui = u. All workers at firm i receive
the same payoff u and in equilibrium this must be the same across all firms. Note that individual
9
firms choose employment Li and employment contracts (αi , βi ) while perfect competition in the
labor market determines the total payoff u of the workers. Compensation is therefore a product of
both internal (moral hazard) and external (market competition) considerations.
The first step in the construction of a competitive labor market equilibrium is the demand
for labor by an individual firm i. In the present context, the usual “price-taking” assumption is
replaced by a “payoff-taking” assumption where each firm believes that it can hire as many workers
as it wants as long as it offers the market payoff u. The demand for labor by firm i will therefore
be a function
Li (u | pMi , Ai , Ki , r, σi2 ),
(22)
where changes in u correspond to movements along the curve while changes in the parameters
(pMi , Ai , Ki , r, σi2 ) shift the curve.
The various costs and benefits of employment Li given that the firm offers optimal incentives
are evident from (17), (19), and (20). First, an increase in employment increases expected output
and revenue. Second, an increase in employment leads to weaker incentives and therefore less
effort. This reduces the risk premium and the cost of effort in (17), which lowers the necessary
expected payment wi . Nevertheless, the expected wage bill wi Li rises. The optimal employment
level balances these tradeoffs. Substituting βi from (20) into (19),8
Πi =
p2 A4 f (Ki )4
2 Mi i
Li − uLi − ρi Ki .
2 Ai f (Ki )2 + rσi2 Li
(23)
Proposition 1 The demand for labor by firm i is given by
2
Li u | pMi , Ai , Ki , r, σi =


√
A2i f (Ki )2 [ pMi Ai f (Ki )− 2u ]
√
rσi2 2u

0
if pMi Ai f (Ki ) ≥
√
2u
(24)
otherwise.
At positive employment levels, employment is increasing in the price pMi , total factor productivity
Ai , and capital Ki and decreasing in the CARA coefficient r, the marginal variance σi2 , and the
outside option u.
In the expression (17) for the expected wage wi , the outside option u represents the external
or market cost of employment. At positive employment levels, the demand for labor is therefore
downward-sloping in u (with u on the vertical axis and Li on the horizontal). The internal costs
of employment are the cost of effort and the risk premium which must be compensated to satisfy
8
A function g is increasing if x > y implies g(x) > g(y) and nondecreasing if x > y implies g(x) ≥ g(y).
10
the participation constraint. An increase in the CARA r or the marginal variance σi2 raises the
internal cost of employment which shifts the demand for labor to the left. An increase in the price
pMi or total factor productivity Ai shifts demand to the right because they increase the value of
employment in terms of expected output. Firms with greater capital Ki have a greater demand for
labor because capital and labor are complements in the production function (3).
The aggregate demand for workers is the horizontal sum
d
L
n
X
n
n
n
2 n
u {pMi }i=1 , {Ai }i=1 , {Ki }i=1 , r, {σi }i=1 =
Li u | pMi , Ai , Ki , r, σi2 .
(25)
i=1
Note that the short-run aggregate demand for labor does not depend on capital costs {ρi } but the
long-run aggregate demand for labor will. The final ingredient is the supply Ls (u) of workers which
we assume is increasing in u.9 A competitive labor market equilibrium is a u ≥ 0 such that supply
equals demand
Ls (u) = Ld (u | {pMi }, {Ai }, {Ki }, r, {σi2 }).
(26)
In Figure 1 below, we depict the aggregate demand for labor in the special case where there are 10
identical firms who compete in the same product market with the same production function, with
p = 10, A = 10, f (K) = 5, r = 10, and σ = 1. From the expression (24) for optimal employment,
the aggregate demand curve crosses the u axis when u = (1/2)A2 p2 f (K)2 and then asymptotes to
the L axis. Given the basic shape of aggregate demand, a unique equilibrium will exist for any
√
well-behaved supply curve. Figure 1 depicts the unique equilibrium when Ls (u) = 10 u.10
L
7000
6000
5000
4000
3000
2000
1000
20 000
40 000
60 000
80 000
100 000
u
120 000
Figure 1: Equilibrium in the Labor Market.
9
We are therefore implicitly assuming that workers have different valuations of leisure or different employment
opportunities outside the specific labor market which is our focus. Workers are otherwise identical.
10
For this particular supply curve, the equilibrium has a unique closed-form solution but it is not sufficiently
interesting to report it here.
11
When markets are perfectly competitive, the choices and parameters which are specific to one
firm have little or no effect on equilibrium prices. In our model, this means that the choices
(αi , βi , Li ) and parameters (Ai , Ki , σi2 ) that are specific to firm i have a negligible impact on pMi
and u. For example, an increase in productivity Ai or employment Li at a single firm i will have
no effect on the equilibrium payoff u of the workers or the competitive price pMi .
3.2
The Optimal Incentive
Substituting the optimal employment level (24) into the previous expression (20) for the optimal
incentive given employment,
Proposition 2 Given positive employment, the optimal incentive is given by
√
βi =
2u
,
Ai f (Ki )
(27)
which is increasing in the outside option u and decreasing in total factor productivity Ai and the
capital stock Ki .
The optimal incentive βi depends on Ai and Ki , which are specific to firm i, as well as the
workers’ equilibrium payoff u which is the same for all firms and determined by supply and demand
in the labor market. Comparing the optimal incentive given employment (20) with the optimal
incentive (27), we observe that taking into account the endogeneity of employment has completely
changed the nature of incentives. Previously, an increase in pMi , Ai , or Ki resulted in stronger
incentives because these parameters make effort more valuable to the firm and an increase in Ai or
Ki increases the sensitivity of effort to incentives. These results were consistent with the standard
model (21). After substituting the optimal employment level, the optimal incentive (27) no longer
depends on pMi and is decreasing in Ai and Ki which seems counterintuitive. But the previous result
(20) was predicated on a fixed level of employment. When employment is endogenous, an increase
in pMi , Ai , or Ki has the further effect, in addition to those above, of increasing employment which
makes effort harder to measure and increases the cost of incentives in terms of the risk premium. In
the case of pMi these two effects exactly offset, while in the case of Ai or Ki the second (negative)
effect operating through employment dominates.
In most of the literature, which usually assumes a single agent, the outside option u has no
effect on incentives and is typically normalized to zero. Its only effect is on the fixed component
of pay (αi in our model) which determines the division of the total surplus between the principal
and agent. But when employment is endogenous, an increase in u increases the external cost of
12
employment, reduces employment, and makes effort easier to measure. This reduces the cost of
incentives in terms of the risk premium and leads to stronger incentives.
The evidence for this prediction is sketchy and indirect but nevertheless encouraging. Bishop
(1987, p. 548) finds that wages are more responsive to productivity in larger labor markets. His
explanation is that “workers have a greater range of choices” which is broadly consistent with ours.
In their survey of R&D engineers, Zenger and Lazzarini (2004) asked respondents to evaluate the
degree to which their current job was “the only option available” and found that this was negatively
correlated with pay mix (the ratio of bonuses to fixed pay) as well as perceived incentive intensity.
Although neither of these findings directly addresses our prediction that incentives are increasing
in the workers’ outside option, they are at least consistent with it.
3.3
The Holmström and Milgrom (1994) Framework
Some of the stylized facts we want to explain involve relationships between endogenous variables;
e.g., the stylized facts that wages are increasing in profit and firm size. To generate empirical
predictions of this sort we use the framework in Holmström and Milgrom (1994). Let x and y
be random variables and Cov(x, y) denote their covariance. We say that a vector x of random
variables is associated if Cov[g(x), h(x)] ≥ 0 for all real-valued nondecreasing functions g and h.
It follows that the random variables x have pairwise nonnegative covariances because g and h can
be taken to be the appropriate projection maps. Intuitively, the concept of association captures a
strong form of correlation which includes independence as a special case. According to Theorem
2(iv) in Holmström and Milgrom (1994), if x is a vector of associated random variables and g is a
vector of nondecreasing real-valued functions then (g(x), x) is associated. The vector (g(x), x) will
therefore exhibit nonnegative pairwise covariances.
3.4
Incentives and Firm Size
In the present context, we make the following assumption.
Assumption 1 Given any values for the CARA coefficient r and the payoff u of the workers, let
T s be the set of vectors
Pis = pMi , Ai , Ki , 1/σi2
(28)
where the optimal employment level (24) and expected operating profit
pMi Ai f (Ki )Ei − wi Li
13
(29)
are nonnegative. We assume the vector Pis of random variables is associated on T s .
The expression in (29) is expected revenue minus expected variable cost (the expected wage
bill) and the requirement that it be nonnegative is the standard short-run shutdown condition
which appears in elementary textbooks. Given any values for r and u, T s is defined as the set of
all parameter configurations Pis such that employment is nonnegative and the firm does not shut
down. The assumption is then that the random variables Pis are associated on the set T s .
This assumption implies that the parameters Pis are positively correlated on T s . This is a
crucial assumption so we discuss its plausibility. In our model, pMi is the perfectly competitive
price determined by supply and demand in product market Mi where firm i operates. Since the
parameters (Ai , Ki , σi2 ) specific to firm i have no effect on pMi , there cannot be any correlation
stemming from an influence of (Ai , Ki , σi2 ) upon pMi .
We now consider Ai and σi2 , which characterize the production technology of firm i. One
possibility, which seems quite plausible, is that these parameters are independent.11 Another is
that firms’ production technologies can be ranked vertically, so that firms with superior technologies
have high total factor productivity Ai and low marginal variance σi2 . In this case, Ai and 1/σi2 will
be positively correlated. What the assumption rules out is negative correlation between Ai and
1/σi2 , which would imply that high-productivity firms tend to be high-volatility firms.
Finally, we consider the capital stock Ki which is fixed in the short run. In the next section
we show that the optimal long-run capital stock Ki is nondecreasing in the vector Pil of long-run
parameters which includes (pMi , Ai , 1/σi2 ). If the parameters Pil exhibit persistence across time then
l
Kit , which was chosen optimally in the past based on Pi,t−1
, should be positively correlated with
2 12 In that case, P s will be associated, where the strength of
l , which includes (p
Pi,t
Mi,t , Ai,t , 1/σi,t ).
i
the positive correlations depends on the age of the capital stock and the extent of the persistence
of the remaining parameters across time.
To apply Theorem 2(iv) in Holmström and Milgrom (1994), all the endogenous variables need
to be nondecreasing in the parameters. Since the optimal incentive (27) is decreasing in total factor
11
Holmström and Milgrom (1994) assume independence in a similar context.
l
l
Formally, assume that (Pi,t−1
, Pi,t
) is associated which is a strong form of persistence. In the next section we show
l
l
l
that Ki is nondecreasing in the long-run parameters, so (Ki (Pi,t−1
), Pi,t−1
, Pi,t
) is associated. Since any subvector
s
of an associated vector of random variables is associated, Pi,t is associated.
12
14
productivity Ai and the capital stock Ki , we define the reciprocal as follows13
β̃i (Pis ) =
1
Ai f (Ki )
.
= √
s
βi (Pi )
2u
(30)
We consider β̃i and the other endogenous variables to be functions solely of the parameters Pis
and not (r, u) because we consider variations in the exogenous and endogenous variables with (r, u)
held fixed. Since optimal employment (24) and the reciprocal β̃i of the optimal incentive are
nondecreasing in Pis on the region T s ,
Proposition 3 Given any values for r and u, the vector (β̃i (Pis ), Li (Pis ), Pis ) is associated on T s .
This result applies purely at the individual firm level and does not involve any equilibrium
considerations. Given any value r for the CARA coefficient and any value u for the workers’
payoff (equilibrium or not), the exogenous variables Pis and endogenous variables β̃i and Li should
exhibit nonnegative pairwise covariances. In particular, the incentive βi should have nonpositive
covariances with both capital Ki and employment Li . Since this result is set in the short run when
capital stocks are fixed, these predictions are valid for cross-sectional data or panel data with a
relatively short time horizon. Intuitively, firms that have a high value for one parameter will tend
to have high values for all the parameters. Since employment is increasing in the parameters, such
firms will tend to be large employers with large capital stocks which offer weak incentives because
effort is hard to measure in large firms.
As far as we know, the only other paper which explicitly models the relationship between
employment and incentives is Rauh (2014). In that paper, we consider the effects of specialization
and division of labor in the context of Holmström and Milgrom (1987) and show that the relationship
between employment and incentives can be positive or negative (Corollary 1). We also show that
the expected wage is increasing in employment under certain conditions (Theorem 2).
All the available evidence supports the prediction that incentives are weaker in large employers.
Rasmusen and Zenger (1990) find a positive relationship between wages and job tenure and that
this relationship is stronger for large firms. Moreover, regressions of weekly earnings on tenure,
outside experience, and education have a larger residual variance for small firms. These findings are
consistent with the hypothesis in Garen (1985) and the present paper that performance is harder to
13
For notational simplicity and to conform with the formal statement of Theorem 2(iv) in Holmström and Milgrom
(1994), we consider β̃i to be a function of Pis despite the fact that it does not depend on pMi or σi2 . We follow this
practice throughout the paper. Note that β̃i is increasing in the parameters Ai and Ki which actually appear as
arguments and nondecreasing in the rest.
15
measure in large firms, which therefore compensate on the basis of easily observed characteristics
such as education and seniority, while small firms directly reward performance. Bishop (1987)
provides direct evidence that wages increase with productivity for small but not for large firms.
Zenger and Lazzarini (2004) find that the pay mix and subjectively perceived incentive intensity
decline with firm size, albeit in a small survey of R&D engineers. Zenger and Marshall (2000) focus
exclusively on group-based rewards (as in this section) and find that incentives decline with firm
size for profit-sharing plans which operate at the firm level and that incentives decline with unit
size for gain-sharing plans which operate at the unit level (i.e., division, facility, department, work
group, and small teams).
The above evidence concerns the relationship between incentives and employment. Garen (1994)
and Schaefer (1998) find evidence consistent with our other prediction that incentives decline with
firm size as measured by capital. To explain their empirical findings, both authors develop agency
models where the size of the firm (the book value of assets or market value) makes effort harder to
measure in the same way that greater employment makes effort harder to measure in our model.
In our model, the effect of capital on incentives operates through a different channel; i.e., through
its complementarity with employment.
3.5
The Risk-Reward Tradeoff
The risk-reward tradeoff present in the optimal incentive with exogenous employment (20) is no
longer evident in the optimal incentive with endogenous employment (27). Specifically, βi in (27)
does not depend on the marginal variance σi2 . When employment is exogenous, an increase in σi2
makes effort harder to measure and leads to weaker incentives as is evident from (20). But when
employment is endogenous, an increase in σi2 reduces employment which makes effort easier to
measure. In our model, these two effects cancel. This result is a special case of Proposition 3 in
Liang, Rajan, and Ray (2008), where the variance of the productivity shock is assumed to take
the form CLγi σ 2 , where C is a positive constant, Li is employment, γ is a positive constant, and
σ 2 is the base variance. In our model, this structure arises because the shock i is the sum of the
individual shocks ij added by each worker.
Taking the log on both sides of (27),
√
ln βi = ln 2u − ln Ai − ln f (Ki ),
(31)
where Ai and Ki are specific to firm i and in equilibrium u is the same across firms. Given
16
perfect data, the expression (31) should hold exactly and the marginal variance σi2 should not have
any explanatory power for incentives. In practice, we may have observable measures of human
capital such as education, experience, and job tenure, but total factor productivity Ai also captures
unobservable human capital as well as process improvements, specialization and division of labor,
and other technological factors related to productivity which may not be observed. Likewise, we
may have a measure of the capital stock Ki but not the productivity of capital f (Ki ) which appears
in (31). In that case, a proxy for σi2 may appear to have explanatory power in a regression with the
incentive βi as the dependent variable when σi2 is correlated with Ai and f (Ki ). In fact, we may
find a positive relationship between incentives and risk as predicted in Proposition 3. On the other
hand, if 1/σi2 is independent of the other parameters, the correlation is weak, or capital stocks are
sufficiently old, we may not observe any relationship at the firm level although we may observe one
at the market level as shown below.
The absence of a risk-reward tradeoff at the firm level is consistent with the mixed nature of the
evidence as surveyed in Prendergast (2002a). Indeed, the most consistent finding in the empirical
literatures on franchising and sharecropping is that the risk-reward relationship is a positive one.
After his survey of the empirical literature, Prendergast (2002a) develops a single-agent moral
hazard model to explain such a positive relationship. In his model, when the principal is uncertain
about which task the agent should perform, she optimally delegates that decision to the agent and
offers incentives tied to output. When there is less uncertainty, the principal simply commands
the agent to perform a particular task and then monitors effort. This can lead to a positive
relationship between incentives and risk. Devaro and Kurtulus (2010) provide evidence in support
of these predictions as well as a more recent survey of the empirical literature.
Other explanations include Prendergast (2002b), which predicts a positive relationship between
incentives and risk when performance measurement is subjective and susceptible to favoritism and
strategic monitoring. Guo and Ou-Yang (2006) show that the risk-reward tradeoff may not obtain
when the agent controls both the mean and the variance of the performance measure. Later we
will see that a positive relationship between incentives and risk emerges in our model in the long
run when capital stocks are endogenous.
3.6
Wages and Expected Wages
Substituting optimal effort (16), the optimal incentive (27), and the optimal employment level (24)
into the previous expression for the expected wage (17),
17
Proposition 4 The expected wage is given by
√
pMi Ai f (Ki ) u
√
+ u,
wi =
2
(32)
which is increasing in the price pMi , total factor productivity Ai , the capital stock Ki , and the
outside option u.
An increase in the price, total factor productivity, or capital leads to greater employment
which makes effort harder to measure. This necessitates an increase in the expected wage as a
compensating differential for the increase in the risk premium. The expected wage is also increasing
in the outside option to satisfy the participation constraint. The model therefore predicts differences
in expected wages across firms due to differences in the parameters (pMi , Ai , Ki ), which correspond
to differences in product market conditions and factors which determine the productivity of labor.
Note that these differences in expected wages will not induce search by workers because the payoff
u is the same across firms.
3.7
The Size-Wage Differential
One of the most durable findings in the empirical labor literature is that wages are positively
correlated with firm size as measured by employment. Once again we can apply the relevant
theorem in Holmström and Milgrom (1994) since wi is nondecreasing in Pis by Proposition 4 and
the parameters Pis are associated by Assumption 1.
Proposition 5 Given r and u, the vector (β̃i (Pis ), Li (Pis ), wi (Pis ), Pis ) is associated on T s .
We can therefore add the expected wage wi to the previous list of exogenous and endogenous
variables which should exhibit nonnegative pairwise covariances. Furthermore,
Lemma 1 If xi is any random variable which is independent from the productivity shock i then
Cov(Ii , xi ) = Cov(wi , xi ).
Proof. Since xi and i are independent,
Cov(Ii , xi ) = E(Ii xi ) − E(Ii )E(xi ) = E[(wi + βi i )xi ] − E(wi )E(xi )
(33)
= E(wi xi + βi i xi ) − E(wi )E(xi ) = E(wi xi ) − E(wi )E(xi )
(34)
= Cov(wi , xi ),
(35)
18
which completes the proof.
All the exogenous and endogenous variables are independent from the productivity shock, so we
can also add the wage Ii to the list of variables which have nonnegative pairwise covariances. Since
the expected wage wi is increasing rather than merely nondecreasing in its arguments, we would
expect positive and significant coefficients on capital Ki and employment Li in cross-sectional
regressions with the wage Ii as the dependent variable. Intuitively, firms that are high in one
parameter will tend to be high in all of them and will therefore tend to be large employers. Since
effort is hard to measure in large firms, they will tend to offer weak incentives but high expected
wages as a compensating differential for risk.
3.7.1
Empirical Evidence
In an influential paper, Brown and Medoff (1989) find a substantial size-wage differential which
cannot be explained by differences in working conditions (e.g., less autonomy in larger firms),
the threat of unionization, labor or product market power, or monitoring costs. They show that
differences in worker quality explain about one-half of that differential while the other explanations
account for little. In this paper we have assumed identical workers and perfect competition in the
product and labor markets. Our model can therefore explain the size-wage differential without
reference to worker quality or market power.
Similarly, Troske (1999) finds that wages are increasing in the capital-labor ratio, the skill
of the workforce (e.g., the percentage of workers with at least a college degree), product market
concentration, and the skill of managers. The prediction that wages should be increasing in the skill
of the workforce comes from matching models such as Kremer (1993), which predict that workers
will be matched with other workers of like skill and paid accordingly. As predicted by efficiency
wage models, Troske finds that wages are negatively correlated with monitoring (the number of
supervisors divided by employment). But only the capital-labor ratio and workforce skill explain
the size-wage differential and both together account for only about half. The other factors have a
direct impact on wages but are uncorrelated with employment; i.e., they have no indirect effect on
wages operating through employment. It is precisely these indirect effects which have the potential
to explain the size-wage differential.
In our model, the parameters that increase the expected wage (32) and do so operating through
employment (24) are the price pMi , total factor productivity Ai , and the capital stock Ki . Now
consider a wage equation which includes employment as an explanatory variable. Since optimal
employment (24) includes Ai and f (Ki ), adding capital Ki and variables that capture measurable
19
human capital should reduce the explanatory power of employment and therefore explain a portion
of the size-wage differential. To explain the remainder we would need to include all important
aspects of total factor productivity and the productivity of capital. This discussion is broadly
consistent with Troske’s findings on the capital-labor ratio and various measures of human capital
and workforce skill. Up to this point, the empirical literature has focused mainly on product
market concentration and other measures of market power with mixed success. In our model firms
are perfectly competitive and it is the price pMi and other variables related to profitability which
contribute to the size-wage differential. We will address this issue further in the context of the
literature on rent-sharing.
3.7.2
Theoretical Explanations
A variety of theoretical explanations have been put forward to explain the size-wage differential. In
this discussion we focus on more recent contributions and contributions which have some overlap
with this paper. Hamermesh (1980) attributes the size-wage differential to the fact that larger
firms tend to be more capital-intensive, which is complementary with worker quality. It follows
that larger firms will employ higher-quality workers and therefore pay higher wages. As discussed
earlier, the capital-labor ratio does indeed explain part of the size-wage differential but differences
in worker quality only explain about one-half of it. Similarly, our model predicts that wages should
be correlated with investments in capital and that some of this effect operates through employment.
The mechanism, however, is different. In our model, workers are identical. An increase in capital
induces an increase in employment because they are complements in production. This increase in
employment leads to an increase in the expected wage because of the increase in the risk premium
which necessitates a compensating differential.
Zábojnı́k and Bernhardt (2001) develop a tournament model which captures some of the ideas
in Hamermesh (1980). In their model, firms with superior production technologies or higher prices
hire more workers. Greater employment translates into more competitive promotion tournaments
and therefore greater investments in human capital. Tournament winners must be paid a premium
to prevent poaching by other firms and since the expected productivity of the winner is increasing
in the size of the tournament, so is the wage that must be paid to winners. In summary, more
profitable firms hire more workers, who make greater investments in human capital, and are paid
higher wages.
Another potential explanation is monopsony power. In a perfectly competitive labor market,
the supply of labor to an individual firm is perfectly elastic and there is no relationship between
20
wages and firm size. But if employers have market power in the labor market they will face an
upward-sloping supply curve and larger firms will have to pay higher wages. As Green, Machin, and
Manning (1996) note, the supply of labor to an individual firm can be less than perfectly elastic for
a variety of reasons. One reason could be search frictions, as in Burdett and Mortensen (1998).14
Green et al. use that model to show that the size-wage differential should be increasing in the
equilibrium level of profit and find evidence in support of that prediction. Note that our model
also features a positive relationship between wages and employment but this is not an upwardsloping supply curve in the traditional monopsony sense. Instead, in our model wages increase with
employment operating through the risk premium.
The class of models most similar to ours is the class of efficiency wage models, especially the
version in Mehta (1998).15 From our perspective, the main insight of this literature is that if
the probability of detecting shirking declines with firm size (or the cost of monitoring increases)
then larger firms will pay higher wages (and monitor less). These models can explain the finding
in Troske (1999) (as well as other papers) that monitoring and wages are negatively correlated.
On the other hand, Troske also finds that monitoring does not explain the size-wage differential
because it is uncorrelated with employment. Finally, the same prediction seems to be at odds with
the evidence that incentives decline with firm size.
Rent-sharing models explain the size-wage differential as follows. Firms with a technological or
product market advantage will tend to be larger and more profitable. If workers are somehow able to
capture some of this surplus then larger firms will pay higher wages. The class of matching models
in Bertola and Garibaldi (2001) can be thought of in this way. In their model, the endogenous
relationship between wages and employment at the firm level is a negative one. What produces the
size-wage differential in aggregate data is the interaction between wages, employment, and firmspecific random shocks. In particular, a positive shock to the workers’ marginal revenue product will
increase both employment and profitability. If the total surplus is split between firms and workers
in a fixed way, then greater expected surplus will translate into higher wages. The prediction that
wages and profits should be positively correlated finds support in Blanchflower, Oswald, and Sanfey
14
Shi (2002) explains the size-wage differential using a combination of labor and product market power in a model
of directed search in both markets. Each worker produces one unit of output, so a firm’s employment level fixes its
capacity. In the product market, consumers can only visit one firm due to high search costs. Firms post prices and
then allocate output randomly among visiting consumers. These consumers are therefore willing to trade off a higher
price in return for a higher probability of obtaining the good. Since that probability is increasing in the employment
level of the firm (i.e., capacity), large firms can charge high prices and are therefore willing to offer high wages. A
size-wage differential obtains when demand is moderate and persists because of search frictions in the labor market.
15
Our model belongs to the class of principal-agent models characterized by a tradeoff between efficiency and
insurance. In contrast, the basic efficiency wage model is isomorphic to the basic principal-agent model characterized
by a tradeoff between efficiency and limited liability rent extraction. See Laffont and Martimort (2002, p. 174).
21
(1996) and Arai (2003) as well as several other studies.
Most of these papers explain the size-wage differential on the basis of external market forces.
The contribution of our paper is to complement the existing theoretical literature by showing that
their results are re-enforced when we look inside the firm to consider the effects on wages which
arise from incentive contracting. One advantage of our approach is that our model can explain a
wide variety of durable empirical phenomena in addition to the size-wage differential.
3.8
Rent-Sharing
We now show that under certain conditions our model implies a nonnegative correlation between
wages and profits, consistent with the empirical rent-sharing literature. Consider the expression
(23) for the expected profit Πi (Pis ) of firm i, where Li is the optimal employment level. We can
apply the envelope theorem to conclude that Πi (Pis ) is increasing in all the parameters Pis except
possibly the capital stock Ki . In the short run, Ki is fixed and not necessarily optimal. If Ki is
less than the optimal capital stock then an increase in Ki will increase Πi but if Ki exceeds the
optimum then an increase in Ki will decrease Πi . Since Πi is not increasing in all the parameters,
we cannot apply the framework in Holmström and Milgrom (1994). Intuitively, the problem is that
firms with excessive capital stocks will offer relatively high wages (32) but will have low expected
profit. If, however, we can assume that T s in Assumption 1 is a lattice and that the vector Pis of
parameters is affiliated then we have the following result.16
Proposition 6 If the vector Pis of parameters is affiliated on T s then given r and u
β̃i (Pis ), Li (Pis ), wi (Pis ), Πi (Pis ), P̃is
(36)
is associated conditional on Ki = K i , where P̃is = (pMi , Ai , 1/σi2 ).
For any fixed value of the capital stock Ki , the endogenous variables (β̃i , Li , wi , Πi ) and the
remaining exogenous variables P̃is will exhibit nonnegative pairwise covariances. This includes, in
particular, the expected wage wi and expected profit Πi . Since wi is increasing in its arguments, we
would expect a positive and significant coefficient on Πi in a regression with wi as the dependent
variable and which includes controls for capital Ki . Adding explanatory variables which capture
16
Assume X is a lattice. A vector x = (x1 , x2 ) of random variables is affiliated on X if it is associated on
every sublattice of X. In particular, if x is affiliated then the subvector x1 is associated for any fixed x2 = x2
because {x | x2 = x2 } is a sublattice. If g is a vector of real-valued functions which are nondecreasing in x1 (but not
necessarily x2 ) then (g(x), x1 ) is associated for any fixed x2 = x2 . See Holmström and Milgrom (1994, p. 980-1) for
further discussion.
22
aspects of prices pMi and total factor productivity Ai may weaken the empirical relationship between
wi and Πi but will not eliminate it as long as the regressors fail to adequately capture these
parameters. Note that the main empirical results in Blanchflower et al. (1996, Table 2) are based
on industry averages of wages and profits. Their main finding that average wages are positively
related to average profit is therefore consistent with the above result. Also note that Blanchflower et
al. obtain this result without controlling for capital or the capital-labor ratio. This is to be expected
when the number of firms with excessive capital stocks is sufficiently small so that controlling for
capital becomes unnecessary.
We now turn to the relationship between wages Ii and profit Π̃i as opposed to the relationship
between their expectations as discussed above. According to Lemma 1, the covariance between the
wage Ii and any random variable xi which is independent of the productivity shock i is the same
as the covariance between the expected wage wi and xi . The problem is that when we consider
the relationship between wages Ii and profit Π̃i , the latter does depend on i . Instead, we have the
following result.
Lemma 2
Cov(Ii , Π̃i ) = Cov(wi , Πi ) + σi2 E [βi (pMi − βi Li )] .
(37)
Π̃i = Πi + i (pMi − βi Li ).
(38)
Proof. From (10),
Since i is independent from all the other parameters,
Cov(Ii , Π̃i ) = E(Ii Π̃i ) − E(Ii )E(Π̃i )
(39)
= E [(wi + βi i )(Πi + i (pMi − βi Li ))] − E(wi )E(Πi )
= E wi Πi + βi i Πi + wi i (pMi − βi Li ) + 2i βi (pMi − βi Li ) − E(wi )E(Πi )
(40)
= E(wi Πi ) − E(wi )E(Πi ) + σi2 E [βi (pMi − βi Li )]
(42)
= Cov(wi , Πi ) + σi2 E [βi (pMi − βi Li )] ,
(43)
(41)
which completes the proof.
Consider a positive shock which increases output by i . From the worker’s perspective, this
increases the wage Ii by βi i . From the firm’s perspective, it increases revenue by pMi i but also
increases costs by Li βi i . For a sufficiently large firm, the overall impact on profit Π̃i will be
negative. A positive productivity shock therefore increases wages Ii but may reduce profit Π̃i ,
23
introducing a negative correlation in the relationship between the two, which is therefore weaker
than the unambiguously positive relationship between expected wages wi and expected profit Πi .
Nevertheless, Arai (2003, Tables 3 and 4) finds a positive relationship between Ii and Π̃i controlling
for the capital-labor ratio. In our model, this occurs when the incentive βi or marginal variance σi2
is sufficiently small or the covariance Cov(wi , Πi ) is sufficiently large.
In traditional rent-sharing models, such findings are explained in terms of bargaining, fairness,
and similar considerations. In our model, rents are shared via incentive contracts which reward
workers based on group performance which is positively correlated with profit. Our model therefore
suggests that a portion of the “rents” obtained by workers in profitable industries can be explained
by performance-related pay, which need not be explicit.
3.9
Common Shocks
We can also use the above approach to incorporate common shocks. For example, consider the case
where total factor productivity Ai = AMi + γi can be decomposed into a common shock AMi which
affects all firms in product market Mi and an idiosyncratic shock γi specific to firm i. Let pMi
be the perfectly competitive price in product market Mi determined by supply and demand. This
will be a function pMi (AMi , ζi ) of the demand parameters ζi as well as the common productivity
shock AMi but it will not depend on parameters specific to firm i such as γi . Unfortunately, pMi is
decreasing in AMi because it shifts the aggregate supply curve in the product market to the right.
This upsets the monotonicity we need. If, however, we can assume that pMi is increasing in ζi and
the new vector of parameters
P̂is = (ζi , AMi , γi , Ki , 1/σi2 )
(44)
is affiliated then the endogenous variables
pMi (P̂is ), β̃i (P̂is ), Li (P̂is ), wi (P̂is ), Πi (P̂is )
(45)
and the remaining exogenous variables (ζi , γi , 1/σi2 ) will be associated for each fixed value for the
capital stock Ki and the common shock AMi . In this case, the relevant regressions would have
not control not only for capital stocks but also include industry and/or sector dummy variables to
control for common shocks.
24
3.10
Labor Market Effects
Up to this point, our analysis has not involved any equilibrium considerations. We now consider
P
the effects of shifts in aggregate demand and supply in the labor market. Let L = ni=1 Li denote
total employment.
Corollary 1
(i) An increase in price pMi for all firms will result in an increase in incentives βi , expected wages
wi , and total employment L.
(ii) An increase in the marginal variance σi2 for all firms or an increase in the CARA coefficient
r will reduce incentives, expected wages, and total employment.
(iii) If the supply of labor shifts to the left, incentives and expected wages will rise while total
employment will fall.
All of these effects can be read from the expressions for optimal employment (24), incentives
(27), and the expected wage (32). For example, consider an increase in price for all firms. While
an increase in the price pMi specific to firms operating in product market Mi may or may not
affect the aggregate demand for labor, depending on the number of such firms, an increase in all
prices surely will. The direct effect of the general price increase is to increase employment (24)
and expected wages (32) with no effect on incentives (27). The indirect effects all follow from the
increase in the aggregate demand for labor, which increases the expected payoff u of the workers
as well as total employment L. The increase in u leads to an increase in incentives and a further
increase in expected wages. Total employment L rises but employment Li at firm i can rise or fall
depending on the specific parameters.17
As we have seen, the optimal incentive (27) does not depend directly on the marginal variance
σi2
so there is no direct relationship between incentives and risk at the firm level. There is, however,
a risk-reward tradeoff at the market level in the sense that an increase in σi2 for all firms leads to
weaker incentives. The only direct effect of an increase in σi2 for all i is to reduce employment Li for
all i. The indirect effects stem from the shift in the aggregate demand for labor to the left, which
reduces u and total employment L. The decrease in u leads to weaker incentives and lower expected
wages at all firms.18 Once again, total employment L falls but the effect on employment Li at firm
17
The increase in u will have feedback effects in all the relevant product markets, shifting market supply curves to
the left, resulting in further increases in pMi for all firms. Assuming stability, the end result will be a higher expected
payoff u for the workers and higher prices pMi for all firms, so the above comparative statics remain valid allowing
for these general equilibrium effects.
18
The rightward shift of the market supply curves in the product markets reinforces these comparative statics.
25
i is ambiguous, depending on the parameters. The only evidence for this prediction seems to be
Garen (1994), who considers the ratio of firm R&D expenditures to book value of assets, averaged
across the relevant industry. As predicted, Garen finds a negative relationship between incentives
and industry risk.19
The final result (iii) makes some novel connections between contract theory and broader social
trends. For example, consider a reduction in the labor force participation rate. This has no direct
effects, but the leftward shift of the labor supply curve leads to an increase in u and a reduction in
total employment L. Not surprisingly, this leads to higher expected wages. More interestingly, it
also leads to stronger incentives as firms substitute effort from the remaining workers to compensate
for lower employment. We are unaware of any evidence on this point.
3.11
Individual Incentives
For simplicity, we have assumed that incentives are tied to group performance qi but there is little
reason to believe that incentives for individual performance would qualitatively change the results.
Consider the signal interpretation of the model, where (5) is deterministic output. The individual
output of worker j at firm i is given by
qij = Ai f (Ki )eij ,
where qi =
P
j qij .
(46)
Assume the only contractible performance measure is a signal yij = qij + j
of individual performance, where j is idiosyncratic noise. What drives the above results is that
the variance of the shock is increasing in employment so that effort becomes harder to measure as
the firm grows. As long as j has that property, we should expect similar results. For example, if
j is i.i.d. normal with mean zero and variance σi2 Li then clearly our results would be completely
unaffected.
4
The Long Run
In the long run, firms are free to adjust their capital stocks. The problem of firm i is to choose
capital Ki ≥ 0 and labor Li ≥ 0 to maximize expected profit (23). The long-run costs and benefits
of capital, given that employment is chosen optimally, are evident from (23). The benefit is that
19
Note that Garen uses this measure of industry risk to proxy individual firm risk. After adding industry dummies,
the coefficient on industry risk becomes insignificant. According to Garen (p. 1189), “This is not surprising given
that much of the variation in industry R&D is explained by aggregate industry dummies.”
26
capital increases expected output as can be seen in the numerator of the first term. The direct cost
of capital is the final term ρi Ki . Since an increase in capital induces an increase in employment,
the indirect costs of capital are the increase in the risk premium (the denominator of the first
term) as well as the external cost of employment uLi . The optimal capital stock balances these
tradeoffs. Unfortunately, the firm’s maximization problem does not have a closed-form solution for
any standard functional form for f (Ki ). We can, however, use lattice programming methods to
achieve the main objective of the paper, which is to obtain comparative statics results that can be
taken to the data.
Proposition 7 Given r and u, assume there exists a subset T l of parameter space
Pil = (pMi , Ai , 1/σi2 , 1/ρi )
(47)
where the problem of firm i has a unique interior solution Ki (Pil ) and Li (Pil ) and expected profit is
nonnegative. Then on this region T l of parameter space, Ki and Li are nondecreasing in each of
the parameters Pil .
The proof is in the appendix. These comparative statics results are intuitively straightforward
given that capital and labor are complements in the production function (3), so a change in some
parameter which increases one will tend to increase the other. An increase in the price pMi or total
factor productivity Ai makes employment and capital more valuable to the firm which leads to
an increase in both. An increase in the marginal variance σi2 or the CARA coefficient r increases
the internal cost of employment in terms of the risk premium. This will reduce employment and
therefore capital. Finally, an increase in the cost of capital ρi will lead to a reduction in capital in
the first instance followed by a complementary reduction in employment.
The more interesting comparative statics results pertain to incentives and expected wages and
profits. Note that the above expressions for employment (24), incentives (27), and the expected
wage (32) remain valid in the long run. For these endogenous variables we have strict comparative
statics results instead of the weak ones in Proposition 7 above. To simplify the statements of the
next two results, we assume the comparative statics results for capital in the above proposition are
strict instead of weak. For example, we assume Ki is increasing rather than nondecreasing in Ai .
Assumption 2 Assume the comparative statics results for capital Ki in Proposition 7 are strict
instead of weak.
To obtain the reciprocal of the long-run incentive, we substitute the optimal capital stock Ki (Pil )
27
into the expression (30) for the reciprocal of the short-run incentive
β̃i (Pil ) =
Ai f (Ki (Pil ))
√
.
2u
(48)
Similarly, the long-run expected wage wi (Pil ) is obtained by substituting Ki (Pil ) into the short-run
expected wage (32) and long-run expected profit Πi (Pil ) by substituting Ki (Pil ) and Li (Pil ) into
short-run expected profit (23). Already we observe some important differences between the short
and long runs. First, the long-run incentive βi now depends directly on the marginal variance σi2
and the CARA coefficient r whereas the short-run incentive did not. This is because capital is
now endogenous and depends on those parameters. Furthermore, all the endogenous variables now
depend on the cost of capital ρi .
Proposition 8 The long-run expected wage wi (Pil ), expected profit Πi (Pil ), and reciprocal of the
incentive β̃i (Pil ), are all increasing in each of the parameters Pil on the region T l .
According to this result, an increase in the CARA coefficient r or the marginal variance σi2
leads to stronger incentives βi and a lower expected wage wi . In the short run there is no direct
relationship between incentives and risk whereas in the long run there is a direct and positive
relationship. The long-run result may seem counterintuitive, especially from a contract theory
perspective grounded in the standard risk-reward tradeoff, but as we have seen there is substantial
evidence to support this prediction.
Intuitively, an increase in r or σi2 with capital held fixed increases the cost of incentives in terms
of the risk premium but also reduces employment which makes effort easier to measure. As we saw
in the previous section, these two effects cancel and there is no effect on the optimal incentive in
the short run. But in the long run capital is endogenous and an increase in r or σi2 reduces capital
which makes effort less valuable to the firm and reduces the sensitivity of effort to incentives. Both
of these effects point towards weaker incentives. But the reduction in capital also leads to a further
complementary decline in employment which again makes effort easier to measure. The second
effect dominates, so the optimal incentive is increasing in r and σi2 in the long run when capital is
endogenous. Finally, the reduction in employment lowers the risk premium which implies a smaller
compensating differential for risk. The expected wage wi therefore declines.
The same logic explains the effects of increases in the price pMi , total factor productivity Ai ,
and the cost of capital ρi . The results for ρi are significant in that we are unaware of any other
theoretical model or empirical evidence which draws a connection between incentive intensity and
the cost of capital. In our model, an increase in ρi leads to a reduction in not only capital but also
28
employment, which makes effort easier to measure and leads to stronger incentives.
Under the assumption that the vector Pil of exogenous parameters in (47) is associated, the
model generates the following long-run empirical predictions.
Proposition 9 If Pil is associated on T l then the vector
β̃i (Pil ), Ki (Pil ), Li (Pil ), wi (Pil ), Πi (Pil ), Pil
(49)
is associated on T l for any given r and u.
We first consider the appropriateness of the assumption that Pil is associated. We focus on
the new parameter ρi , which reflects the cost of borrowing for firm i. The inverse 1/ρi should be
positively correlated with the price pMi , total factor productivity Ai , and the inverse 1/σi2 of the
marginal variance because all of these parameters increase expected profit and therefore reduce
credit risk.
This result strengthens and extends the corresponding short-run result in the previous section.
It strengthens our previous results in the sense that we no longer need to assume affiliation but
rather the weaker assumption of association. In practical terms, it means that we no longer have
to control for capital stocks in the relevant regressions. The result extends previous ones in the
sense that most of the short-run predictions in the previous section are preserved in the long run.
For example, incentives continue to be negatively correlated with firm size as measured either by
employment or capital and the expected wage remains positively correlated with both firm size and
expected profit. Note that Lemmas 1 and 2 continue to hold in the long run, so the covariance
between wages Ii and the other exogenous and endogenous variables will be positive under the same
assumptions as before. The exceptions stem from the fact that capital is now endogenous and the
cost of capital ρi now influences the endogenous variables. In particular, the long run incentive is
positively correlated with the marginal variance σi2 and the endogenous variables are all negatively
correlated with the cost of capital ρi .
5
Conclusion
In this paper we extended the workhorse moral hazard model in Holmström and Milgrom (1987) to
encompass not only incentives but also employment, capital, and the external labor market. The
resulting model is simple, with straightforward explanations for sometimes counterintuitive results,
and has the same wide scope of application as the textbook neoclassical model. It therefore has the
29
potential to serve as a benchmark model for thinking and making predictions about employment,
capital, incentives, wages, and profit. The main assumptions of the model are that effort becomes
harder to measure as the firm gets larger and the exogenous parameters are affiliated. We showed
that the endogenous variables and exogenous parameters can be defined in such a way that the
former are all at least nondecreasing in the latter. An application of Holmström and Milgrom (1994)
allowed us to conclude that the endogenous and exogenous variables should exhibit nonnegative
pairwise covariances in data sets with the appropriate time frame.
The potential of the model to serve as a useful theoretical benchmark is illustrated by its ability
to explain several stylized facts: why incentives decline but wages rise with firm size as measured
by either employment or capital, the mixed nature of the evidence on the risk-reward tradeoff,
and the positive correlation between wages and profits. These predictions are straightforward
consequences of the fact that employment is endogenous and effort becomes harder to measure
as employment grows. In our model, the firm under moral hazard operates within a competitive
labor market which determines the overall payoff of the workers, while incentives and wages are
set within the firm subject to the endogenous participation constraint. This aspect of the model
allows us to make novel predictions about the relationship between incentives and aggregate labor
market conditions. In particular, any labor market phenomenon which increases the workers’ total
payoff (e.g., a decrease in labor force participation) will lead to stronger incentives. When capital
is endogenous, we obtain the counterintuitive but empirically relevant prediction that incentives
are positively related to risk in the long run. Another novel prediction is that incentives should
be positively related to the cost of capital, which makes a potential connection between incentive
contracting and macroeconomic conditions in the form of interest rates.
6
Appendix
Proof of Proposition 7. We apply Theorem 2.3 in Vives (1999, p. 26). We drop the subscript
i throughout the proof. Let X be the set of all (K, L) ∈ (0, ∞) × (0, ∞). Note that X is a
lattice and T l is a partially ordered subset of R4 which is nonempty by assumption. We denote
derivatives using subscripts. The following cross-partials show that expected profit (23) is strictly
30
supermodular on X with strictly increasing differences on X × T l .
ΠKL =
ΠKA =
ΠKpM =
ΠKσ2
ΠLA
ΠLpM
ΠLσ2
A6 p2M f (K)5 f 0 (K) A2 f (K)2 + 3Lrσ 2
>0
[A2 f (K)2 + Lrσ 2 ]3
2A3 Lp2M f (K)3 f 0 (K) A4 f (K)4 + 3A2 Lrσ 2 f (K)2 + 4L2 r2 σ 4
[A2 f (K)2 + Lrσ 2 ]3
2LpM f 0 (K) A6 f (K)5 + 2A4 Lrσ 2 f (K)3
[A2 f (K)2 + Lrσ 2 ]2
2A4 L3 p2M r2 σ 2 f (K)3 f 0 (K)
=−
<0
[A2 f (K)2 + Lrσ 2 ]3
A5 p2M f (K)6 A2 f (K)2 + 3Lrσ 2
=
>0
[A2 f (K)2 + Lrσ 2 ]3
A6 pM f (K)6
=
>0
[A2 f (K)2 + Lrσ 2 ]2
A6 Lp2M rf (K)6
=−
<0
[A2 f (K)2 + Lrσ 2 ]3
>0
>0
ΠLρ = 0, and ΠKρ = −1. From Theorem 2.3 cited above we conclude that K and L are at least
weakly increasing in the parameters.
7
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